In the case of Denmark, they did it through legislation. They made it a requirement that both men and women, if they are in a partnered relationship, take responsibility for early child care. They would take time off work. If not, it would be lost for both parties. What happened was that, over a generation, as more and more men participated in work without the negative consequences that are associated for a lot of men with taking time away from work, they have become one of the most gender-equal places on earth. Men now see that they have a role to play in care at all stages of life.
On May 30th, 2017. See this statement in context.